April  2013, 6(4): 891-908. doi: 10.3934/dcdss.2013.6.891

Intertwining semiclassical solutions to a Schrödinger-Newton system

1. 

Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari

2. 

via Orabona 4, 70125 Bari

3. 

Instituto de Matemticas, Universidad Nacional Autnoma de Mxico

4. 

Circuito Exterior, C.U., 04510 Mxico D.F.

5. 

Dipartimento di Matematica ed Applicazioni

6. 

Universit di Milano-Bicocca

7. 

edificio U5, via R. Cozzi 53, I-20125 Milano

Received  October 2011 Revised  February 2012 Published  December 2012

We study the problem\[\begin{cases}\left( -\varepsilon\mathrm{i}\nabla+A(x)\right) ^{2}u+V(x)u=\varepsilon^{-2}\left( \frac{1}{|x|}\ast|u|^{2}\right) u,\\u\in L^{2}(\mathbb{R}^{3},\mathbb{C}),    \varepsilon\nablau+\mathrm{i}Au\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{3}),\end{cases}\]where $A\colon\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ is an exterior magneticpotential, $V\colon\mathbb{R}^{3}\rightarrow\mathbb{R}$ is an exteriorelectric potential, and $\varepsilon$ is a small positive number. If $A=0$ and$\varepsilon=\hbar$ is Planck's constant this problem is equivalent to theSchr?dinger-Newton equations proposed by Penrose in [23] todescribe his view that quantum state reduction occurs due to somegravitational effect. We assume that $A$ and $V$ are compatible with theaction of a group $G$ of linear isometries of $\mathbb{R}^{3}$. Then, for anygiven homomorphism $\tau:G\rightarrow\mathbb{S}^{1}$ into the unit complexnumbers, we show that there is a combined effect of the symmetries and thepotential $V$ on the number of semiclassical solutions $u:\mathbb{R}^{3}\rightarrow\mathbb{C}$ which satisfy $u(gx)=\tau(g)u(x)$ for all $g\in G$,$x\in\mathbb{R}^{3}$. We also study the concentration behavior of thesesolutions as $\varepsilon 0.$
Citation: Silvia Cingolani, Mnica Clapp, Simone Secchi. Intertwining semiclassical solutions to a Schrödinger-Newton system. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 891-908. doi: 10.3934/dcdss.2013.6.891
References:
[1]

L. Abatangelo and S. Terracini, Solutions to nonlinear Schrödinger equations with singular electromagnetic potential and critical exponent,, J. Fixed Point Theory Appl., 10 (2011), 147. doi: 10.1007/s11784-011-0053-0.

[2]

N. Ackermann, On a periodic Schrödinger equation with nonlocar superlinear part,, Math. Z., 248 (2004), 423. doi: 10.1007/s00209-004-0663-y.

[3]

S. Cingolani and M. Clapp, Intertwining semiclassical bound states to a nonlinear magnetic equation,, Nonlinearity, 22 (2009), 2309. doi: 10.1088/0951-7715/22/9/013.

[4]

S. Cingolani and M. Clapp, Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory,, Commun. Pure Appl. Anal., 9 (2010), 1263. doi: 10.3934/cpaa.2010.9.1263.

[5]

S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation,, Z. Angew. Math. Phys. 63 (2012), 63 (2012), 233. doi: 10.1007/s00033-011-0166-8.

[6]

S. Cingolani, S. Secchi and M. Squassina, Semiclassical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities,, Proc. Roy. Soc. Edinburgh, 140 A (2010), 973. doi: 10.1017/S0308210509000584.

[7]

M. Clapp and D. Puppe, Critical point theory with symmetries,, J. Reine Angew. Math., 418 (1991), 1. doi: 10.1007/BF02566437.

[8]

M. Clapp and A. Szulkin, Multiple solutions to a nonlinear Schrödinger equation with Aharonov-Bohm magnetic potential,, Nonlinear Differ. Equ. Appl. (NoDea), 17 (2010), 229. doi: 10.1007/s00030-009-0051-8.

[9]

T. tom Dieck, "Transformation Groups,", Walter de Gruyter, (1987). doi: 10.1515/9783110858372.312.

[10]

J. Fröhlich and E. Lenzmann, Mean-field limit of quantum Bose gases and nonlinear Hartree equation,, in, (2004), 2003.

[11]

J. Fröhlich, T.-P. Tsai and H.-T. Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation,, Comm. Math. Phys., 225 (2002), 223. doi: 10.1007/s002200100579.

[12]

R. Harrison, I. Moroz and K. P. Tod, A numerical study of the Schrödinger-Newton equations,, Nonlinearity, 16 (2003), 101. doi: 10.1088/0951-7715/16/1/307.

[13]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,, Stud. Appl. Math., 57 (1977), 93.

[14]

E. H. Lieb and M. Loss, "Analysis,", Graduate Studies in Math. 14, 14 (1997).

[15]

P.-L. Lions, The concentration-compacteness principle in the calculus of variations. The locally compact case,, Ann. Inst. Henry Poincaré, 1 (1984), 109.

[16]

P.-L. Lions, The Choquard equation and related questions,, Nonlinear Anal. T.M.A., 4 (1980), 1063. doi: 10.1016/0362-546X(80)90016-4.

[17]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Rational Mech. Anal., 195 (2010), 455. doi: 10.1007/s00205-008-0208-3.

[18]

I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations,, Topology of the Universe Conference (Cleveland, 15 (1998), 2733. doi: 10.1088/0264-9381/15/9/019.

[19]

I. M. Moroz and P. Tod, An analytical approach to the Schrödinger-Newton equations,, Nonlinearity, 12 (1999), 201. doi: 10.1088/0951-7715/12/2/002.

[20]

M. Nolasco, Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential,, Commun. Pure Appl. Anal., 9 (2010), 1411. doi: 10.3934/cpaa.2010.9.1411.

[21]

R. Palais, The principle of symmetric criticallity,, Comm. Math. Phys., 69 (1979), 19.

[22]

R. Penrose, On gravity's role in quantum state reduction,, Gen. Rel. Grav., 28 (1996), 581. doi: 10.1007/BF02105068.

[23]

R. Penrose, Quantum computation, entanglement and state reduction,, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927. doi: 10.1098/rsta.1998.0256.

[24]

R. Penrose, "The Road to Reality. A Complete Guide to the Laws of the Universe,", Alfred A. Knopf Inc., (2005).

[25]

S. Secchi, A note on Schrödinger-Newton systems with decaying electric potential,, Nonlinear Analysis, 72 (2010), 3842. doi: 10.1016/j.na.2010.01.021.

[26]

P. Tod, The ground state energy of the Schrödinger-Newton equation,, Physics Letters A, 280 (2001), 173. doi: 10.1016/S0375-9601(01)00059-7.

[27]

J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equation,, J. Math. Phys., 50 (2009). doi: 10.1063/1.3060169.

[28]

M. Willem, "Minimax Theorems,", PNLDE 24, 24 (1996). doi: 10.1007/978-1-4612-4146-1.

show all references

References:
[1]

L. Abatangelo and S. Terracini, Solutions to nonlinear Schrödinger equations with singular electromagnetic potential and critical exponent,, J. Fixed Point Theory Appl., 10 (2011), 147. doi: 10.1007/s11784-011-0053-0.

[2]

N. Ackermann, On a periodic Schrödinger equation with nonlocar superlinear part,, Math. Z., 248 (2004), 423. doi: 10.1007/s00209-004-0663-y.

[3]

S. Cingolani and M. Clapp, Intertwining semiclassical bound states to a nonlinear magnetic equation,, Nonlinearity, 22 (2009), 2309. doi: 10.1088/0951-7715/22/9/013.

[4]

S. Cingolani and M. Clapp, Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory,, Commun. Pure Appl. Anal., 9 (2010), 1263. doi: 10.3934/cpaa.2010.9.1263.

[5]

S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation,, Z. Angew. Math. Phys. 63 (2012), 63 (2012), 233. doi: 10.1007/s00033-011-0166-8.

[6]

S. Cingolani, S. Secchi and M. Squassina, Semiclassical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities,, Proc. Roy. Soc. Edinburgh, 140 A (2010), 973. doi: 10.1017/S0308210509000584.

[7]

M. Clapp and D. Puppe, Critical point theory with symmetries,, J. Reine Angew. Math., 418 (1991), 1. doi: 10.1007/BF02566437.

[8]

M. Clapp and A. Szulkin, Multiple solutions to a nonlinear Schrödinger equation with Aharonov-Bohm magnetic potential,, Nonlinear Differ. Equ. Appl. (NoDea), 17 (2010), 229. doi: 10.1007/s00030-009-0051-8.

[9]

T. tom Dieck, "Transformation Groups,", Walter de Gruyter, (1987). doi: 10.1515/9783110858372.312.

[10]

J. Fröhlich and E. Lenzmann, Mean-field limit of quantum Bose gases and nonlinear Hartree equation,, in, (2004), 2003.

[11]

J. Fröhlich, T.-P. Tsai and H.-T. Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation,, Comm. Math. Phys., 225 (2002), 223. doi: 10.1007/s002200100579.

[12]

R. Harrison, I. Moroz and K. P. Tod, A numerical study of the Schrödinger-Newton equations,, Nonlinearity, 16 (2003), 101. doi: 10.1088/0951-7715/16/1/307.

[13]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,, Stud. Appl. Math., 57 (1977), 93.

[14]

E. H. Lieb and M. Loss, "Analysis,", Graduate Studies in Math. 14, 14 (1997).

[15]

P.-L. Lions, The concentration-compacteness principle in the calculus of variations. The locally compact case,, Ann. Inst. Henry Poincaré, 1 (1984), 109.

[16]

P.-L. Lions, The Choquard equation and related questions,, Nonlinear Anal. T.M.A., 4 (1980), 1063. doi: 10.1016/0362-546X(80)90016-4.

[17]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Rational Mech. Anal., 195 (2010), 455. doi: 10.1007/s00205-008-0208-3.

[18]

I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations,, Topology of the Universe Conference (Cleveland, 15 (1998), 2733. doi: 10.1088/0264-9381/15/9/019.

[19]

I. M. Moroz and P. Tod, An analytical approach to the Schrödinger-Newton equations,, Nonlinearity, 12 (1999), 201. doi: 10.1088/0951-7715/12/2/002.

[20]

M. Nolasco, Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential,, Commun. Pure Appl. Anal., 9 (2010), 1411. doi: 10.3934/cpaa.2010.9.1411.

[21]

R. Palais, The principle of symmetric criticallity,, Comm. Math. Phys., 69 (1979), 19.

[22]

R. Penrose, On gravity's role in quantum state reduction,, Gen. Rel. Grav., 28 (1996), 581. doi: 10.1007/BF02105068.

[23]

R. Penrose, Quantum computation, entanglement and state reduction,, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927. doi: 10.1098/rsta.1998.0256.

[24]

R. Penrose, "The Road to Reality. A Complete Guide to the Laws of the Universe,", Alfred A. Knopf Inc., (2005).

[25]

S. Secchi, A note on Schrödinger-Newton systems with decaying electric potential,, Nonlinear Analysis, 72 (2010), 3842. doi: 10.1016/j.na.2010.01.021.

[26]

P. Tod, The ground state energy of the Schrödinger-Newton equation,, Physics Letters A, 280 (2001), 173. doi: 10.1016/S0375-9601(01)00059-7.

[27]

J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equation,, J. Math. Phys., 50 (2009). doi: 10.1063/1.3060169.

[28]

M. Willem, "Minimax Theorems,", PNLDE 24, 24 (1996). doi: 10.1007/978-1-4612-4146-1.

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